Gauss’s Lemma is not a horrible combinatorial calculation

I’m not really sure why I’ve never seen this argument before, but it’s much cleaner and easier to digest than the arguments I’m used to that make Gauss’s Lemma look like it’s not part of algebra at all.

Definition. A polynomial with integer coefficients is primitive if its coefficients are relatively prime.

Lemma. (Gauss) The product of primitive polynomials is primitive.

Proof. Notice first that a polynomial is primitive iff no prime divides all its coefficients, iff it is not the zero polynomial (mod p) for any prime p.

Suppose two polynomials f and g are primitive. Then for each prime p neither f nor g is the zero element in \mathbb{F}_p[x]. Since this ring is an integral domain, it follows that f g is not the zero element either. As this holds for all primes p, f g is primitive.

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How to get GAP4 to describe a group

This is a command that’s tough to Google even if you know it exists.  To get GAP to describe a group with a common, human-readable name, you can use the StructureDescription command, as follows:

gap> StructureDescription(Group([(1,3), (1,2,3,4)]));

"D8"

gap> StructureDescription(Group([(1,3), (1,2,3)]));

"S3"

Documentation for theStructureDescription command is here.

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1988 IMO Problem 6

Problem 6 of the 1988 International Math Olympiad reads:

Let a and b be positive integers such that a b + 1 divides a^2 + b^2.  Show that \displaystyle\frac{a^2 + b^2}{a b + 1} is the square of an integer.

There’s a writeup of a solution at the Art of Problem Solving wiki, but I found it incredibly sketchy and difficult to read and the article was locked, so here’s what’s hopefully an easier-to-read exposition of the same solution:

In other words, we’d like to show that any solution in positive integers to the equation a^2 + b^2 = c(a b + 1) has c a perfect square.

First, suppose we have a solution with a = b.  Then a^2 + 1 divides 2 a^2. Applying the Euclidean algorithm,

\gcd(a^2 + 1, 2 a^2) = \gcd(a^2 - 1, a^2 + 1) = \gcd(2, a^2 - 1) \in \{1, 2\}

so we must have a^2 + 1 = \{1, 2\}.  Since a is positive, we must have a=1.  In this case, a^2 + 1 = 2a^2 = 2 so c = 1 which is a perfect square.  So we may assume in what follows that a \neq b.

Fix a positive integer c for which the equation has solutions in the positive integers, and let (a, b, c) be such a solution with the minimum possible value of a.  Note that by symmetry we must have a < b, since otherwise (b, a, c) is a solution with a smaller first coordinate.

Rewrite the equation as

b^2 - c a b + (a^2 - c) = 0

and notice that this says that b is a root of the quadratic equation

x^2 - c a x + (a^2 - c) = 0

Let r denote the other root.  Then b + r = c a and b r = a^2 - c.  It follows that r = c a - b is an integer, and that

a r < br = a^2 - c \leq a^2

So r < a.  This must mean that r \leq 0, as otherwise the solution (r, b, c) to the original equation would contradict the minimality of a.

Now

(b+1)(r + 1) = b r + (b + r) + 1 = (a^2 - c) + c a + 1 = a^2 + c(a - 1) + 1 \geq 1

Since b + 1 is positive, we must have r + 1 positive as well.  Since r is an integer, we must have r = 0, so

a^2 - c = b r = 0

or in other words c = a^2.

 

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The Poisson Summation Formula

There are various conventions for normalizing the Fourier transform. For the purposes of this post, let’s write

[\mathscr{F}(f)](s) := \displaystyle\int_{-\infty}^{\infty} e^{i s t} f(t) \, dt

The Poisson Summation Formula concerns the sum

\displaystyle\sum_{n=-\infty}^{\infty} [\mathscr{F}(f)](n)

Where on Earth did that come from? Well, without regard to convergence issues, we have

\displaystyle\sum_{n=-\infty}^{\infty} [\mathscr{F}(f)](n) = \sum_{n=-\infty}^{\infty} \int_{-\infty}^{\infty} e^{i n t} f(t) \, dt = \int_{-\infty}^{\infty} \left[\sum_{n=-\infty}^{\infty} e^{i n t}\right] f(t) \, dt

So we see that we’re just transforming our function with respect to the kernel

\displaystyle\sum_{n=-\infty}^{\infty} e^{i n t}

Problematically, this doesn’t converge to a function. However, it does converge to a distribution. Let’s take a look at how it looks for a few values of n:

poisson

Figure: \displaystyle\sum_{n=-N}^N e^{i n t} for N from 0 to 5.

As illustrated by the figure, the sequence is converging to a Dirac Comb, a periodic sequence of Dirac deltas.  From here the Poisson summation formula should be clear.  Alternatively, we could have started by (formally) working out the Fourier series for the Dirac comb, and we’d quickly arrive at the Poisson summation formula.

 

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Elliptic integrals, I

This has tripped me up in the past, so let’s write it down.

Imagine we have an ellipse

\displaystyle\frac{x^2}{a^2} + \displaystyle\frac{y^2}{b^2} = 1

Let’s say we have some parametrization (x(t), y(t)) of the ellipse and we want to convert it into a unit-speed parametrization. We can do this by composing with the inverse of the function s(t) that tells us how much arclength our parametrization has traced out on [0, t], which is given by

s(t) = \displaystyle\int_0^t \sqrt{x'(\tau)^2 + y'(\tau)^2} \, d\tau

Differentiating the equation cutting out our ellipse, we have

\displaystyle\frac{2}{a^2} \; x \, x' + \displaystyle\frac{2}{b^2} \; y \, y' = 0

Let’s solve for y’ so that we can eliminate it from the integral:

y' = \displaystyle \frac{-b^2}{a^2} \, \frac{x}{y} \, x'

Squaring,

(y')^2 = \displaystyle \frac{b^4}{a^4} \, \frac{x^2}{y^2} \, (x')^2

Plugging this in, we have

s(t) = \displaystyle\int_0^t \sqrt{x'(\tau)^2\left[1 + \frac{b^4}{a^4} \, \frac{x^2}{y^2}\right]} \, d\tau = \displaystyle\int_0^t |x'(\tau)| \, \sqrt{1 + \frac{b^4}{a^4} \, \frac{x^2}{y^2}} \; d\tau

Let’s eliminate y from the equation as well. From the defining equation of the ellipse, we have

\displaystyle y^2 = b^2\left(1 - \frac{x^2}{a^2}\right)

Plugging this in, we get

s(t) = \displaystyle\int_0^t |x'(\tau)| \, \sqrt{1 + \frac{b^4}{a^4} \left[ \frac{x^2}{b^2\left(1 - \displaystyle\frac{x^2}{a^2}\right)}\right]} \; d\tau

Now let’s assume that our parametrization has x(\tau) = 0 and x'(\tau) > 0 for small values of \tau — for instance, we could start the parametrization from the top of the ellipse and go counter-clockwise. Then, for x(t) in the second quadrant, we can write

s(t) = \displaystyle\int_0^{x(t)} \sqrt{1 + \frac{b^4}{a^4} \left[ \frac{x^2}{b^2\left(1 - \displaystyle\frac{x^2}{a^2}\right)}\right]} \; dx

= \displaystyle\int_0^{x(t)} \, \sqrt{1 + \frac{b^2 x^2}{a^4 - a^2 x^2}} \; dx

= \displaystyle\int_0^{x(t)} \, \sqrt{1 + \frac{b^2 x^2}{a^4 - a^2 x^2}} \; dx

= \displaystyle\int_0^{x(t)} \, \sqrt{\frac{a^4 - (a^2 - b^2)x^2}{a^2(a^2 - x^2)}} \; dx

= \displaystyle\int_0^{x(t)} \, \frac{\sqrt{a^2 - \left(1 - \displaystyle\frac{b^2}{a^2}\right)x^2}}{\sqrt{a^2 - x^2}} \; dx

Putting k^2 := 1 - \displaystyle\frac{b^2}{a^2}, this becomes

s(t) = \displaystyle\int_0^{x(t)} \frac{\sqrt{a^2 - k^2 x^2}}{\sqrt{a^2 - x^2}} \; dx

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n-fold integral of the natural logarithm

The n-fold integral of the natural logarithm is given by

\frac{1}{n!} z^n \log z - \frac{H_n}{n!}  z^n + C(z)

where H_n is the n-th harmonic number and C(z) is any polynomial of degree at most n-1.

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On the Gamma function

I want to say a bit about how the Gamma function can be characterized, because I’m not a huge fan of the ways I’ve seen in print.

Let’s make this quick.  I want to know about functions γ such that:

  1. γ(1) = 1,
  2. γ satisfies the functional equation* γ(t+1) = t γ(t), and
  3. γ is a meromorphic function on the complex plane*.

Note that conditions (1) and (2) together mean that γ(n) = (n-1)! for positive integers n.

We know that one such function exists, namely the classical Gamma function.  For positive real x we may define

\Gamma(x) := \displaystyle\int_0^\infty t^{x-1} e^{-t} \, dt

and this function has an analytic continuation, which we’ll also denote Γ, that’s a meromorphic function on the complex plane with poles at the nonpositive integers.

Suppose γ is another function satisfying (1) – (3), and consider the function

r(t) = \displaystyle\frac{\gamma(t)}{\Gamma(t)}

Then r is a meromorphic function with r(1) = γ(1)/Γ(1) = 1/1 = 1 and

r(t+1) = \displaystyle\frac{\gamma(t+1)}{\Gamma(t+1)} = \displaystyle\frac{t \gamma(t)}{t \Gamma(t)} = \displaystyle\frac{\gamma(t)}{\Gamma(t)} = r(t)

Conversely, if r is any singly-periodic meromorphic function with period 1 and r(1) = 1, and we let γ(t) := r(t) Γ(t), then γ clearly satisfies (1) – (3).  So, given that we already know about the classical Γ function, the problem of classifying every function satisfying (1) – (3) reduces to the problem of classifying these functions r.

Taking the quotient of the complex plane by the translation sending z to z+1 gives us a cylinder, so equivalently we’re looking for meromorphic functions on this cylinder.  This cylinder is isomorphic (a.k.a. biholomorphic, a.k.a. conformally equivalent) to the punctured complex plane.

Unfortunately the noncompactness of the cylinder is a serious problem if we want to describe its function field.  I’m not an expert here, but my understanding is that the situation looks like this:

  • The holomorphic periodic functions on the cylinder are just given by Fourier series.
  • The meromorphic ones with finitely many poles are just ratios of the holomorphic ones.
  • The meromorphic ones with infinitely many poles, though, aren’t so easy to describe.

One way people try to deal with this situation is by putting some arbitrary analytic condition on the functions that limits exposure to the third class of functions.  This is in essence what the Wielandt characterization of the Gamma function is doing.

For more on the function field of the cylinder, see:


*It’s probably worth saying a couple of things for people who aren’t familiar with the subject.  These are the sorts of things that bothered me when I was first learning complex analysis and which, while sort of appearing in texts, never seem to be highlighted to the extent that they deserve.

First, notice that if we try to plug in t=0 into this functional equation, we get

1 = γ(1) = 0 γ(0),

which we cannot solve for γ(0).  So what do we mean when we say that γ satisfies this functional equation?  Well, we mean that it’s satisfied whenever both t and t+1 are in the domain of γ.  We see in particular that t=0 cannot be in the domain of any function γ satisfying conditions (1)-(3).

This appears at first to open another can of worms, since if we’re allowed to throw points out of the domain of γ at will we could simply look at every pair of points not satisfying the functional equation and throw out one or the other of them at random.

But we’re considering a meromorphic function on the plane.  Such a function is defined at every point in the plane, minus some countable discrete set.  (“Countable” is redundant here — every uncountable set of points in the plane fails to be discrete.)  In fact, by the Riemann removable singularities theorem, we can further insist that such a function is only undefined where it “has to be” (i.e., where the function approaches infinity in magnitude near a point).  To be quite technical, we generally think not of a particular meromorphic function but rather of an equivalence classes of meromorphic functions under the relation f~g if f(z) = g(z) at every point z in the domain of both functions.

Anyway, the point is that everything’s fine — we won’t have any analytic pathologies creeping in.

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